Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem only known to be true $r\leq 3$, or when $\nu(H)=1$ and 5$. An equivalent formulation in $r$-edge coloring a graph $G$ independence $\alpha(G)$, there exists $(r-1)\alpha(G)$ monochromatic connected subgra...

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition...

Given a graph G and a natural number k, the k graph product of G = (V,E) is the graph with vertex set V . For every two vertices x = (x1, . . . , xk) and y = (y1, . . . , yk) in V , an edge is placed according to a predefined rule. Graph products are a basic combinatorial object, widely studied and used in different areas such as hardness of approximation, information theory, etc. We study grap...

A famous conjecture (usually called Ryser’s conjecture), appeared in the Ph.D thesis of his student, J. R. Henderson [9], states that for an r-uniform r-partite hypergraph H, the inequality τ(H) ≤ (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem [16], and in the case of r = 3, which was proved by Aharoni in 2001 [2]...

The best degree-based upper bound for the spectral radius is due to Liu and Weng. This paper begins by demonstrating that a (forgotten) upper bound for the spectral radius dating from 1983 is equivalent to their much more recent bound. This bound is then used to compare lower bounds for the clique number. A series of line graph degree-based upper bounds for the Q-index is then proposed and comp...

An equitable k-coloring of a graph G is proper such that the sizes any two color classes differ by at most one. (q,r)-tree-coloring an q-coloring subgraph induced each class forest maximum degree r. Let strong vertex r-arboricity G, denoted var≡(G), be minimum p has for every q≥p. The values va1≡(Kn,n) were investigated Tao and Lin Wu, Zhang, Li where exact found in some special cases. In this ...

A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete m-partite digraph. We describe results (theorems and algorithms) on directed walks in semicomplete mpartite digraphs including some recent results concerning tournaments.

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition...